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Calculus

# Some Finite Series

Natural numbers: $$n$$, $$k$$

Number of terms of a series: $$n$$

1. Sum of the first $$n$$ natural numbers
$$1 + 2 + 3 + \ldots + n =$$ $${\large\frac{{n\left( {n + 1} \right)}}{2}\normalsize}$$
2. Sum of the first $$n$$ even natural numbers
$$2 + 4 + 6 + \ldots + 2n =$$ $${n\left( {n + 1} \right)}$$
3. Sum of the first $$n$$ odd natural numbers
$$1 + 3 + 5 + \ldots$$ $$+ \left( {2n – 1} \right) =$$ $${n^2}$$
4. Sum of $$n$$ natural numbers starting from $$k$$
$$k + \left( {k + 1} \right) + \left( {k + 2} \right) + \ldots$$ $$+ \left( {k + n – 1} \right) =$$ $${\large\frac{{n\left( {2k + n – 1} \right)}}{2}\normalsize}$$
5. Sum of the squares of the first $$n$$ natural numbers
$${1^2} + {2^2} + {3^2} + \ldots + {n^2} =$$ $${\large\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\normalsize}$$
6. Sum of the cubes of the first $$n$$ natural numbers
$${1^3} + {2^3} + {3^3} + \ldots + {n^3} =$$ $${\left[ {{\large\frac{{n\left( {n + 1} \right)}}{2}}\normalsize} \right]^2}$$
7. Sum of the squares of the first $$n$$ odd natural numbers
$${1^2} + {3^2} + {5^2} + \ldots$$ $$+ {\left( {2n – 1} \right)^2} =$$ $${\large\frac{{n\left( {4{n^2} – 1} \right)}}{3}\normalsize}$$
8. Sum of the cubes of the first $$n$$ odd natural numbers
$${1^3} + {3^3} + {5^3} + \ldots$$ $$+ {\left( {2n – 1} \right)^3} =$$ $${n^2}\left( {2{n^2} – 1} \right)$$
9. $${\large\frac{1}{{1 \cdot 2}}\normalsize} + {\large\frac{1}{{2 \cdot 3}}\normalsize} + {\large\frac{1}{{3 \cdot 4}}\normalsize} + \ldots$$ $$+ {\large\frac{1}{{n \left( {n + 1} \right)}}\normalsize} =$$ $${\large\frac{n}{{n + 1}}\normalsize} =$$ $$1 – {\large\frac{1}{{n + 1}}\normalsize}$$
10. $${\large\frac{1}{{1 \cdot 2 \cdot 3}}\normalsize} + {\large\frac{1}{{2 \cdot 3 \cdot 4}}\normalsize} + {\large\frac{1}{{3 \cdot 4 \cdot 5}}\normalsize} + \ldots$$ $$+ {\large\frac{1}{{n\left( {n + 1} \right)\left( {n + 2} \right)}}\normalsize} =$$ $${\large\frac{1}{2}\normalsize}\left[ {{\large\frac{1}{2}\normalsize} – {\large\frac{1}{{\left( {n + 1} \right)\left( {n + 2} \right)}}}\normalsize} \right]$$